LMFDB - Embedded newform 539.4.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,4,Mod(1,539)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("539.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	539.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.8020294931$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.44399$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.44399 q^{2} -8.26395 q^{3} +11.7491 q^{4} +22.0150 q^{5} -36.7249 q^{6} +16.6609 q^{8} +41.2928 q^{9} +97.8345 q^{10} +11.0000 q^{11} -97.0937 q^{12} +51.5769 q^{13} -181.931 q^{15} -19.9519 q^{16} +26.5590 q^{17} +183.505 q^{18} -99.6432 q^{19} +258.656 q^{20} +48.8839 q^{22} +28.1455 q^{23} -137.684 q^{24} +359.660 q^{25} +229.207 q^{26} -118.115 q^{27} -43.9369 q^{29} -808.499 q^{30} +83.8402 q^{31} -221.953 q^{32} -90.9034 q^{33} +118.028 q^{34} +485.152 q^{36} +306.353 q^{37} -442.814 q^{38} -426.228 q^{39} +366.789 q^{40} -200.991 q^{41} -13.7546 q^{43} +129.240 q^{44} +909.062 q^{45} +125.079 q^{46} +266.533 q^{47} +164.881 q^{48} +1598.33 q^{50} -219.482 q^{51} +605.980 q^{52} +308.867 q^{53} -524.903 q^{54} +242.165 q^{55} +823.446 q^{57} -195.255 q^{58} +622.446 q^{59} -2137.52 q^{60} +87.3303 q^{61} +372.585 q^{62} -826.742 q^{64} +1135.46 q^{65} -403.974 q^{66} +608.395 q^{67} +312.044 q^{68} -232.593 q^{69} -464.926 q^{71} +687.974 q^{72} +255.407 q^{73} +1361.43 q^{74} -2972.21 q^{75} -1170.72 q^{76} -1894.16 q^{78} +261.237 q^{79} -439.240 q^{80} -138.809 q^{81} -893.204 q^{82} -953.986 q^{83} +584.696 q^{85} -61.1255 q^{86} +363.092 q^{87} +183.269 q^{88} +839.910 q^{89} +4039.86 q^{90} +330.684 q^{92} -692.851 q^{93} +1184.47 q^{94} -2193.64 q^{95} +1834.21 q^{96} +349.146 q^{97} +454.221 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} - 2 q^{3} + 45 q^{4} + 24 q^{5} - 4 q^{6} + 57 q^{8} + 63 q^{9} + 10 q^{10} + 55 q^{11} - 24 q^{12} + 50 q^{13} - 146 q^{15} + 433 q^{16} - 222 q^{17} + 245 q^{18} - 160 q^{19} + 430 q^{20}+ \cdots + 693 q^{99}+O(q^{100}) $ 5 * q + q^2 - 2 * q^3 + 45 * q^4 + 24 * q^5 - 4 * q^6 + 57 * q^8 + 63 * q^9 + 10 * q^10 + 55 * q^11 - 24 * q^12 + 50 * q^13 - 146 * q^15 + 433 * q^16 - 222 * q^17 + 245 * q^18 - 160 * q^19 + 430 * q^20 + 11 * q^22 + 54 * q^23 + 220 * q^24 + 125 * q^25 + 1026 * q^26 - 110 * q^27 + 14 * q^29 - 1080 * q^30 + 34 * q^31 - 583 * q^32 - 22 * q^33 + 750 * q^34 - 411 * q^36 + 1044 * q^37 + 156 * q^38 - 124 * q^39 - 1158 * q^40 + 114 * q^41 + 672 * q^43 + 495 * q^44 + 530 * q^45 - 1224 * q^46 + 292 * q^47 + 1652 * q^48 + 1143 * q^50 + 768 * q^51 + 914 * q^52 - 710 * q^53 - 1608 * q^54 + 264 * q^55 + 2012 * q^57 - 810 * q^58 - 270 * q^59 - 3068 * q^60 - 138 * q^61 + 480 * q^62 + 3001 * q^64 - 196 * q^65 - 44 * q^66 + 1942 * q^67 - 3130 * q^68 + 1306 * q^69 - 278 * q^71 + 3565 * q^72 + 338 * q^73 + 462 * q^74 - 3960 * q^75 + 1332 * q^76 + 280 * q^78 + 576 * q^79 + 1602 * q^80 - 1439 * q^81 - 1386 * q^82 - 1644 * q^83 + 360 * q^85 + 2020 * q^86 + 1956 * q^87 + 627 * q^88 + 3656 * q^89 + 6162 * q^90 - 748 * q^92 - 1442 * q^93 - 976 * q^94 - 1276 * q^95 + 4508 * q^96 - 692 * q^97 + 693 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.44399	1.57119	0.785594	−	0.618742i	$-0.212357\pi$
$2$	4.44399	1.57119	0.785594	+	0.618742i	$0.212357\pi$
$3$	−8.26395	−1.59040	−0.795199	−	0.606349i	$-0.792633\pi$
$3$	−8.26395	−1.59040	−0.795199	+	0.606349i	$0.792633\pi$
$4$	11.7491	1.46863
$5$	22.0150	1.96908	0.984541	−	0.175155i	$-0.0560428\pi$
$5$	22.0150	1.96908	0.984541	+	0.175155i	$0.0560428\pi$
$6$	−36.7249	−2.49881
$7$	0	0
$7$	0	0
$8$	16.6609	0.736313
$9$	41.2928	1.52936
$10$	97.8345	3.09380
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−97.0937	−2.33571
$13$	51.5769	1.10037	0.550186	−	0.835042i	$-0.314557\pi$
$13$	51.5769	1.10037	0.550186	+	0.835042i	$0.314557\pi$
$14$	0	0
$15$	−181.931	−3.13162
$16$	−19.9519	−0.311748
$17$	26.5590	0.378912	0.189456	−	0.981889i	$-0.439328\pi$
$17$	26.5590	0.378912	0.189456	+	0.981889i	$0.439328\pi$
$18$	183.505	2.40292
$19$	−99.6432	−1.20314	−0.601571	−	0.798819i	$-0.705458\pi$
$19$	−99.6432	−1.20314	−0.601571	+	0.798819i	$0.705458\pi$
$20$	258.656	2.89186
$21$	0	0
$22$	48.8839	0.473731
$23$	28.1455	0.255163	0.127582	−	0.991828i	$-0.459279\pi$
$23$	28.1455	0.255163	0.127582	+	0.991828i	$0.459279\pi$
$24$	−137.684	−1.17103
$25$	359.660	2.87728
$26$	229.207	1.72889
$27$	−118.115	−0.841899
$28$	0	0
$29$	−43.9369	−0.281340	−0.140670	−	0.990057i	$-0.544926\pi$
$29$	−43.9369	−0.281340	−0.140670	+	0.990057i	$0.544926\pi$
$30$	−808.499	−4.92037
$31$	83.8402	0.485747	0.242873	−	0.970058i	$-0.421910\pi$
$31$	83.8402	0.485747	0.242873	+	0.970058i	$0.421910\pi$
$32$	−221.953	−1.22613
$33$	−90.9034	−0.479523
$34$	118.028	0.595342
$35$	0	0
$36$	485.152	2.24608
$37$	306.353	1.36119	0.680596	−	0.732659i	$-0.261721\pi$
$37$	306.353	1.36119	0.680596	+	0.732659i	$0.261721\pi$
$38$	−442.814	−1.89036
$39$	−426.228	−1.75003
$40$	366.789	1.44986
$41$	−200.991	−0.765599	−0.382800	−	0.923831i	$-0.625040\pi$
$41$	−200.991	−0.765599	−0.382800	+	0.923831i	$0.625040\pi$
$42$	0	0
$43$	−13.7546	−0.0487805	−0.0243903	−	0.999703i	$-0.507764\pi$
$43$	−13.7546	−0.0487805	−0.0243903	+	0.999703i	$0.507764\pi$
$44$	129.240	0.442810
$45$	909.062	3.01144
$46$	125.079	0.400909
$47$	266.533	0.827189	0.413594	−	0.910461i	$-0.364273\pi$
$47$	266.533	0.827189	0.413594	+	0.910461i	$0.364273\pi$
$48$	164.881	0.495803
$49$	0	0
$50$	1598.33	4.52075
$51$	−219.482	−0.602621
$52$	605.980	1.61605
$53$	308.867	0.800493	0.400247	−	0.916407i	$-0.368924\pi$
$53$	308.867	0.800493	0.400247	+	0.916407i	$0.368924\pi$
$54$	−524.903	−1.32278
$55$	242.165	0.593700
$56$	0	0
$57$	823.446	1.91348
$58$	−195.255	−0.442039
$59$	622.446	1.37348	0.686742	−	0.726901i	$-0.259040\pi$
$59$	622.446	1.37348	0.686742	+	0.726901i	$0.259040\pi$
$60$	−2137.52	−4.59921
$61$	87.3303	0.183303	0.0916516	−	0.995791i	$-0.470785\pi$
$61$	87.3303	0.183303	0.0916516	+	0.995791i	$0.470785\pi$
$62$	372.585	0.763200
$63$	0	0
$64$	−826.742	−1.61473
$65$	1135.46	2.16672
$66$	−403.974	−0.753421
$67$	608.395	1.10936	0.554681	−	0.832063i	$-0.312840\pi$
$67$	608.395	1.10936	0.554681	+	0.832063i	$0.312840\pi$
$68$	312.044	0.556483
$69$	−232.593	−0.405811
$70$	0	0
$71$	−464.926	−0.777135	−0.388567	−	0.921420i	$-0.627030\pi$
$71$	−464.926	−0.777135	−0.388567	+	0.921420i	$0.627030\pi$
$72$	687.974	1.12609
$73$	255.407	0.409495	0.204747	−	0.978815i	$-0.434363\pi$
$73$	255.407	0.409495	0.204747	+	0.978815i	$0.434363\pi$
$74$	1361.43	2.13869
$75$	−2972.21	−4.57602
$76$	−1170.72	−1.76698
$77$	0	0
$78$	−1894.16	−2.74963
$79$	261.237	0.372043	0.186022	−	0.982546i	$-0.440441\pi$
$79$	261.237	0.372043	0.186022	+	0.982546i	$0.440441\pi$
$80$	−439.240	−0.613857
$81$	−138.809	−0.190410
$82$	−893.204	−1.20290
$83$	−953.986	−1.26161	−0.630804	−	0.775942i	$-0.717275\pi$
$83$	−953.986	−1.26161	−0.630804	+	0.775942i	$0.717275\pi$
$84$	0	0
$85$	584.696	0.746109
$86$	−61.1255	−0.0766434
$87$	363.092	0.447443
$88$	183.269	0.222007
$89$	839.910	1.00034	0.500170	−	0.865927i	$-0.333271\pi$
$89$	839.910	1.00034	0.500170	+	0.865927i	$0.333271\pi$
$90$	4039.86	4.73154
$91$	0	0
$92$	330.684	0.374741
$93$	−692.851	−0.772530
$94$	1184.47	1.29967
$95$	−2193.64	−2.36909
$96$	1834.21	1.95003
$97$	349.146	0.365468	0.182734	−	0.983162i	$-0.441505\pi$
$97$	349.146	0.365468	0.182734	+	0.983162i	$0.441505\pi$
$98$	0	0
$99$	454.221	0.461121
$100$	4225.68	4.22568
$101$	−1492.44	−1.47033	−0.735163	−	0.677890i	$-0.762894\pi$
$101$	−1492.44	−1.47033	−0.735163	+	0.677890i	$0.762894\pi$
$102$	−975.377	−0.946831
$103$	−558.687	−0.534457	−0.267228	−	0.963633i	$-0.586108\pi$
$103$	−558.687	−0.534457	−0.267228	+	0.963633i	$0.586108\pi$
$104$	859.314	0.810218
$105$	0	0
$106$	1372.60	1.25773
$107$	−694.047	−0.627066	−0.313533	−	0.949577i	$-0.601513\pi$
$107$	−694.047	−0.627066	−0.313533	+	0.949577i	$0.601513\pi$
$108$	−1387.74	−1.23644
$109$	−341.005	−0.299654	−0.149827	−	0.988712i	$-0.547872\pi$
$109$	−341.005	−0.299654	−0.149827	+	0.988712i	$0.547872\pi$
$110$	1076.18	0.932815
$111$	−2531.68	−2.16484
$112$	0	0
$113$	−990.910	−0.824929	−0.412464	−	0.910974i	$-0.635332\pi$
$113$	−990.910	−0.824929	−0.412464	+	0.910974i	$0.635332\pi$
$114$	3659.39	3.00643
$115$	619.624	0.502437
$116$	−516.218	−0.413186
$117$	2129.75	1.68287
$118$	2766.15	2.15800
$119$	0	0
$120$	−3031.12	−2.30585
$121$	121.000	0.0909091
$122$	388.095	0.288004
$123$	1660.98	1.21761
$124$	985.045	0.713384
$125$	5166.05	3.69652
$126$	0	0
$127$	−666.090	−0.465401	−0.232700	−	0.972548i	$-0.574756\pi$
$127$	−666.090	−0.465401	−0.232700	+	0.972548i	$0.574756\pi$
$128$	−1898.41	−1.31092
$129$	113.668	0.0775804
$130$	5046.00	3.40433
$131$	−30.4356	−0.0202990	−0.0101495	−	0.999948i	$-0.503231\pi$
$131$	−30.4356	−0.0202990	−0.0101495	+	0.999948i	$0.503231\pi$
$132$	−1068.03	−0.704244
$133$	0	0
$134$	2703.70	1.74302
$135$	−2600.31	−1.65777
$136$	442.496	0.278998
$137$	−2810.25	−1.75252	−0.876262	−	0.481836i	$-0.839970\pi$
$137$	−2810.25	−1.75252	−0.876262	+	0.481836i	$0.839970\pi$
$138$	−1033.64	−0.637605
$139$	−3110.49	−1.89804	−0.949021	−	0.315212i	$-0.897924\pi$
$139$	−3110.49	−1.89804	−0.949021	+	0.315212i	$0.897924\pi$
$140$	0	0
$141$	−2202.62	−1.31556
$142$	−2066.13	−1.22103
$143$	567.345	0.331775
$144$	−823.869	−0.476776
$145$	−967.270	−0.553982
$146$	1135.03	0.643394
$147$	0	0
$148$	3599.36	1.99909
$149$	1916.92	1.05396	0.526979	−	0.849878i	$-0.323325\pi$
$149$	1916.92	1.05396	0.526979	+	0.849878i	$0.323325\pi$
$150$	−13208.5	−7.18980
$151$	−2289.28	−1.23377	−0.616883	−	0.787055i	$-0.711605\pi$
$151$	−2289.28	−1.23377	−0.616883	+	0.787055i	$0.711605\pi$
$152$	−1660.14	−0.885889
$153$	1096.70	0.579494
$154$	0	0
$155$	1845.74	0.956475
$156$	−5007.79	−2.57015
$157$	−280.036	−0.142352	−0.0711762	−	0.997464i	$-0.522675\pi$
$157$	−280.036	−0.142352	−0.0711762	+	0.997464i	$0.522675\pi$
$158$	1160.93	0.584550
$159$	−2552.46	−1.27310
$160$	−4886.29	−2.41435
$161$	0	0
$162$	−616.865	−0.299170
$163$	−866.571	−0.416411	−0.208206	−	0.978085i	$-0.566762\pi$
$163$	−866.571	−0.416411	−0.208206	+	0.978085i	$0.566762\pi$
$164$	−2361.46	−1.12439
$165$	−2001.24	−0.944220
$166$	−4239.51	−1.98223
$167$	1965.18	0.910600	0.455300	−	0.890338i	$-0.349532\pi$
$167$	1965.18	0.910600	0.455300	+	0.890338i	$0.349532\pi$
$168$	0	0
$169$	463.173	0.210820
$170$	2598.39	1.17228
$171$	−4114.55	−1.84004
$172$	−161.604	−0.0716407
$173$	−3956.88	−1.73894	−0.869469	−	0.493988i	$-0.835539\pi$
$173$	−3956.88	−1.73894	−0.869469	+	0.493988i	$0.835539\pi$
$174$	1613.58	0.703018
$175$	0	0
$176$	−219.471	−0.0939956
$177$	−5143.86	−2.18439
$178$	3732.55	1.57172
$179$	−3143.58	−1.31264	−0.656318	−	0.754484i	$-0.727887\pi$
$179$	−3143.58	−1.31264	−0.656318	+	0.754484i	$0.727887\pi$
$180$	10680.6	4.42271
$181$	−683.772	−0.280798	−0.140399	−	0.990095i	$-0.544838\pi$
$181$	−683.772	−0.280798	−0.140399	+	0.990095i	$0.544838\pi$
$182$	0	0
$183$	−721.693	−0.291525
$184$	468.929	0.187880
$185$	6744.36	2.68030
$186$	−3079.03	−1.21379
$187$	292.149	0.114246
$188$	3131.52	1.21484
$189$	0	0
$190$	−9748.54	−3.72228
$191$	2739.68	1.03789	0.518944	−	0.854809i	$-0.326325\pi$
$191$	2739.68	1.03789	0.518944	+	0.854809i	$0.326325\pi$
$192$	6832.15	2.56806
$193$	2651.93	0.989067	0.494534	−	0.869159i	$-0.335339\pi$
$193$	2651.93	0.989067	0.494534	+	0.869159i	$0.335339\pi$
$194$	1551.60	0.574219
$195$	−9383.42	−3.44595
$196$	0	0
$197$	−1879.52	−0.679749	−0.339874	−	0.940471i	$-0.610385\pi$
$197$	−1879.52	−0.679749	−0.339874	+	0.940471i	$0.610385\pi$
$198$	2018.56	0.724507
$199$	3119.39	1.11119	0.555597	−	0.831452i	$-0.312490\pi$
$199$	3119.39	1.11119	0.555597	+	0.831452i	$0.312490\pi$
$200$	5992.25	2.11858
$201$	−5027.75	−1.76433
$202$	−6632.37	−2.31016
$203$	0	0
$204$	−2578.71	−0.885029
$205$	−4424.82	−1.50753
$206$	−2482.80	−0.839733
$207$	1162.21	0.390237
$208$	−1029.05	−0.343039
$209$	−1096.08	−0.362761
$210$	0	0
$211$	−520.718	−0.169894	−0.0849472	−	0.996385i	$-0.527072\pi$
$211$	−520.718	−0.169894	−0.0849472	+	0.996385i	$0.527072\pi$
$212$	3628.90	1.17563
$213$	3842.12	1.23595
$214$	−3084.34	−0.985238
$215$	−302.808	−0.0960528
$216$	−1967.90	−0.619901
$217$	0	0
$218$	−1515.42	−0.470814
$219$	−2110.67	−0.651260
$220$	2845.21	0.871929
$221$	1369.83	0.416944
$222$	−11250.8	−3.40137
$223$	−2101.08	−0.630935	−0.315467	−	0.948936i	$-0.602161\pi$
$223$	−2101.08	−0.630935	−0.315467	+	0.948936i	$0.602161\pi$
$224$	0	0
$225$	14851.4	4.40041
$226$	−4403.60	−1.29612
$227$	6051.96	1.76953	0.884764	−	0.466040i	$-0.154320\pi$
$227$	6051.96	1.76953	0.884764	+	0.466040i	$0.154320\pi$
$228$	9674.73	2.81020
$229$	2995.73	0.864470	0.432235	−	0.901761i	$-0.357725\pi$
$229$	2995.73	0.864470	0.432235	+	0.901761i	$0.357725\pi$
$230$	2753.61	0.789424
$231$	0	0
$232$	−732.026	−0.207155
$233$	−65.3656	−0.0183787	−0.00918936	−	0.999958i	$-0.502925\pi$
$233$	−65.3656	−0.0183787	−0.00918936	+	0.999958i	$0.502925\pi$
$234$	9464.61	2.64411
$235$	5867.73	1.62880
$236$	7313.17	2.01715
$237$	−2158.85	−0.591697
$238$	0	0
$239$	−1102.33	−0.298343	−0.149171	−	0.988811i	$-0.547661\pi$
$239$	−1102.33	−0.298343	−0.149171	+	0.988811i	$0.547661\pi$
$240$	3629.86	0.976277
$241$	−5297.43	−1.41592	−0.707962	−	0.706250i	$-0.750385\pi$
$241$	−5297.43	−1.41592	−0.707962	+	0.706250i	$0.750385\pi$
$242$	537.723	0.142835
$243$	4336.22	1.14473
$244$	1026.05	0.269205
$245$	0	0
$246$	7381.39	1.91309
$247$	−5139.28	−1.32391
$248$	1396.85	0.357661
$249$	7883.69	2.00646
$250$	22957.9	5.80793
$251$	−177.964	−0.0447530	−0.0223765	−	0.999750i	$-0.507123\pi$
$251$	−177.964	−0.0447530	−0.0223765	+	0.999750i	$0.507123\pi$
$252$	0	0
$253$	309.601	0.0769346
$254$	−2960.10	−0.731232
$255$	−4831.90	−1.18661
$256$	−1822.59	−0.444969
$257$	−3496.69	−0.848707	−0.424354	−	0.905497i	$-0.639499\pi$
$257$	−3496.69	−0.848707	−0.424354	+	0.905497i	$0.639499\pi$
$258$	505.138	0.121893
$259$	0	0
$260$	13340.7	3.18212
$261$	−1814.28	−0.430272
$262$	−135.256	−0.0318936
$263$	5747.94	1.34766	0.673828	−	0.738889i	$-0.264649\pi$
$263$	5747.94	1.34766	0.673828	+	0.738889i	$0.264649\pi$
$264$	−1514.53	−0.353079
$265$	6799.71	1.57624
$266$	0	0
$267$	−6940.97	−1.59094
$268$	7148.08	1.62925
$269$	−235.217	−0.0533140	−0.0266570	−	0.999645i	$-0.508486\pi$
$269$	−235.217	−0.0533140	−0.0266570	+	0.999645i	$0.508486\pi$
$270$	−11555.7	−2.60467
$271$	1179.58	0.264406	0.132203	−	0.991223i	$-0.457795\pi$
$271$	1179.58	0.264406	0.132203	+	0.991223i	$0.457795\pi$
$272$	−529.902	−0.118125
$273$	0	0
$274$	−12488.7	−2.75354
$275$	3956.26	0.867533
$276$	−2732.76	−0.595988
$277$	−3638.98	−0.789331	−0.394666	−	0.918825i	$-0.629140\pi$
$277$	−3638.98	−0.789331	−0.394666	+	0.918825i	$0.629140\pi$
$278$	−13823.0	−2.98218
$279$	3462.00	0.742883
$280$	0	0
$281$	3236.81	0.687160	0.343580	−	0.939123i	$-0.388360\pi$
$281$	3236.81	0.687160	0.343580	+	0.939123i	$0.388360\pi$
$282$	−9788.42	−2.06699
$283$	−8303.78	−1.74420	−0.872100	−	0.489328i	$-0.837242\pi$
$283$	−8303.78	−1.74420	−0.872100	+	0.489328i	$0.837242\pi$
$284$	−5462.45	−1.14133
$285$	18128.2	3.76779
$286$	2521.28	0.521281
$287$	0	0
$288$	−9165.06	−1.87520
$289$	−4207.62	−0.856426
$290$	−4298.54	−0.870411
$291$	−2885.32	−0.581239
$292$	3000.80	0.601398
$293$	1894.16	0.377672	0.188836	−	0.982009i	$-0.439529\pi$
$293$	1894.16	0.377672	0.188836	+	0.982009i	$0.439529\pi$
$294$	0	0
$295$	13703.2	2.70450
$296$	5104.10	1.00226
$297$	−1299.27	−0.253842
$298$	8518.76	1.65597
$299$	1451.66	0.280775
$300$	−34920.8	−6.72050
$301$	0	0
$302$	−10173.5	−1.93848
$303$	12333.4	2.33840
$304$	1988.07	0.375077
$305$	1922.58	0.360939
$306$	4873.71	0.910495
$307$	−6596.30	−1.22629	−0.613144	−	0.789971i	$-0.710096\pi$
$307$	−6596.30	−1.22629	−0.613144	+	0.789971i	$0.710096\pi$
$308$	0	0
$309$	4616.96	0.849999
$310$	8202.47	1.50280
$311$	5242.26	0.955824	0.477912	−	0.878408i	$-0.341394\pi$
$311$	5242.26	0.955824	0.477912	+	0.878408i	$0.341394\pi$
$312$	−7101.33	−1.28857
$313$	−5338.75	−0.964103	−0.482051	−	0.876143i	$-0.660108\pi$
$313$	−5338.75	−0.964103	−0.482051	+	0.876143i	$0.660108\pi$
$314$	−1244.48	−0.223662
$315$	0	0
$316$	3069.29	0.546395
$317$	−5807.21	−1.02891	−0.514456	−	0.857517i	$-0.672006\pi$
$317$	−5807.21	−1.02891	−0.514456	+	0.857517i	$0.672006\pi$
$318$	−11343.1	−2.00028
$319$	−483.306	−0.0848273
$320$	−18200.7	−3.17953
$321$	5735.56	0.997283
$322$	0	0
$323$	−2646.42	−0.455885
$324$	−1630.87	−0.279642
$325$	18550.1	3.16608
$326$	−3851.03	−0.654261
$327$	2818.04	0.476570
$328$	−3348.69	−0.563720
$329$	0	0
$330$	−8893.49	−1.48355
$331$	−1366.51	−0.226919	−0.113460	−	0.993543i	$-0.536193\pi$
$331$	−1366.51	−0.226919	−0.113460	+	0.993543i	$0.536193\pi$
$332$	−11208.4	−1.85284
$333$	12650.2	2.08176
$334$	8733.25	1.43072
$335$	13393.8	2.18443
$336$	0	0
$337$	−3363.75	−0.543724	−0.271862	−	0.962336i	$-0.587639\pi$
$337$	−3363.75	−0.543724	−0.271862	+	0.962336i	$0.587639\pi$
$338$	2058.34	0.331239
$339$	8188.83	1.31196
$340$	6869.64	1.09576
$341$	922.242	0.146458
$342$	−18285.0	−2.89106
$343$	0	0
$344$	−229.164	−0.0359177
$345$	−5120.54	−0.799075
$346$	−17584.4	−2.73220
$347$	2984.97	0.461791	0.230896	−	0.972979i	$-0.425834\pi$
$347$	2984.97	0.461791	0.230896	+	0.972979i	$0.425834\pi$
$348$	4265.99	0.657130
$349$	−1286.08	−0.197255	−0.0986276	−	0.995124i	$-0.531445\pi$
$349$	−1286.08	−0.197255	−0.0986276	+	0.995124i	$0.531445\pi$
$350$	0	0
$351$	−6092.01	−0.926403
$352$	−2441.48	−0.369691
$353$	−8417.60	−1.26919	−0.634594	−	0.772846i	$-0.718833\pi$
$353$	−8417.60	−1.26919	−0.634594	+	0.772846i	$0.718833\pi$
$354$	−22859.3	−3.43208
$355$	−10235.3	−1.53024
$356$	9868.16	1.46913
$357$	0	0
$358$	−13970.0	−2.06240
$359$	7483.47	1.10017	0.550087	−	0.835108i	$-0.314595\pi$
$359$	7483.47	1.10017	0.550087	+	0.835108i	$0.314595\pi$
$360$	15145.7	2.21736
$361$	3069.76	0.447553
$362$	−3038.68	−0.441186
$363$	−999.938	−0.144582
$364$	0	0
$365$	5622.79	0.806329
$366$	−3207.20	−0.458041
$367$	−8588.73	−1.22160	−0.610801	−	0.791784i	$-0.709153\pi$
$367$	−8588.73	−1.22160	−0.610801	+	0.791784i	$0.709153\pi$
$368$	−561.556	−0.0795466
$369$	−8299.50	−1.17088
$370$	29971.9	4.21125
$371$	0	0
$372$	−8140.36	−1.13456
$373$	11833.0	1.64260	0.821298	−	0.570500i	$-0.193251\pi$
$373$	11833.0	1.64260	0.821298	+	0.570500i	$0.193251\pi$
$374$	1298.31	0.179502
$375$	−42691.9	−5.87894
$376$	4440.67	0.609070
$377$	−2266.13	−0.309579
$378$	0	0
$379$	5056.39	0.685301	0.342651	−	0.939463i	$-0.388675\pi$
$379$	5056.39	0.685301	0.342651	+	0.939463i	$0.388675\pi$
$380$	−25773.3	−3.47932
$381$	5504.53	0.740172
$382$	12175.1	1.63072
$383$	−6457.09	−0.861467	−0.430733	−	0.902479i	$-0.641745\pi$
$383$	−6457.09	−0.861467	−0.430733	+	0.902479i	$0.641745\pi$
$384$	15688.4	2.08488
$385$	0	0
$386$	11785.1	1.55401
$387$	−567.968	−0.0746032
$388$	4102.14	0.536739
$389$	12444.5	1.62201	0.811004	−	0.585040i	$-0.198921\pi$
$389$	12444.5	1.62201	0.811004	+	0.585040i	$0.198921\pi$
$390$	−41699.9	−5.41424
$391$	747.518	0.0966844
$392$	0	0
$393$	251.518	0.0322835
$394$	−8352.59	−1.06801
$395$	5751.12	0.732584
$396$	5336.68	0.677217
$397$	619.207	0.0782799	0.0391400	−	0.999234i	$-0.487538\pi$
$397$	619.207	0.0782799	0.0391400	+	0.999234i	$0.487538\pi$
$398$	13862.5	1.74589
$399$	0	0
$400$	−7175.90	−0.896987
$401$	9731.89	1.21194	0.605969	−	0.795488i	$-0.292785\pi$
$401$	9731.89	1.21194	0.605969	+	0.795488i	$0.292785\pi$
$402$	−22343.3	−2.77209
$403$	4324.22	0.534502
$404$	−17534.7	−2.15937
$405$	−3055.87	−0.374932
$406$	0	0
$407$	3369.88	0.410415
$408$	−3656.76	−0.443717
$409$	−4621.43	−0.558717	−0.279358	−	0.960187i	$-0.590122\pi$
$409$	−4621.43	−0.558717	−0.279358	+	0.960187i	$0.590122\pi$
$410$	−19663.9	−2.36861
$411$	23223.7	2.78721
$412$	−6564.05	−0.784922
$413$	0	0
$414$	5164.85	0.613137
$415$	−21002.0	−2.48421
$416$	−11447.6	−1.34920
$417$	25704.9	3.01864
$418$	−4870.95	−0.569966
$419$	−186.428	−0.0217365	−0.0108682	−	0.999941i	$-0.503460\pi$
$419$	−186.428	−0.0217365	−0.0108682	+	0.999941i	$0.503460\pi$
$420$	0	0
$421$	2670.29	0.309126	0.154563	−	0.987983i	$-0.450603\pi$
$421$	2670.29	0.309126	0.154563	+	0.987983i	$0.450603\pi$
$422$	−2314.07	−0.266936
$423$	11005.9	1.26507
$424$	5145.99	0.589413
$425$	9552.22	1.09024
$426$	17074.4	1.94192
$427$	0	0
$428$	−8154.40	−0.920930
$429$	−4688.51	−0.527654
$430$	−1345.68	−0.150917
$431$	−12514.9	−1.39866	−0.699328	−	0.714801i	$-0.746517\pi$
$431$	−12514.9	−1.39866	−0.699328	+	0.714801i	$0.746517\pi$
$432$	2356.62	0.262460
$433$	16651.2	1.84805	0.924025	−	0.382332i	$-0.124879\pi$
$433$	16651.2	1.84805	0.924025	+	0.382332i	$0.124879\pi$
$434$	0	0
$435$	7993.47	0.881052
$436$	−4006.49	−0.440083
$437$	−2804.51	−0.306998
$438$	−9379.80	−1.02325
$439$	−6033.38	−0.655940	−0.327970	−	0.944688i	$-0.606364\pi$
$439$	−6033.38	−0.655940	−0.327970	+	0.944688i	$0.606364\pi$
$440$	4034.68	0.437149
$441$	0	0
$442$	6087.51	0.655098
$443$	6320.03	0.677819	0.338910	−	0.940819i	$-0.389942\pi$
$443$	6320.03	0.677819	0.338910	+	0.940819i	$0.389942\pi$
$444$	−29744.9	−3.17935
$445$	18490.6	1.96975
$446$	−9337.17	−0.991318
$447$	−15841.3	−1.67621
$448$	0	0
$449$	−17893.6	−1.88074	−0.940368	−	0.340159i	$-0.889519\pi$
$449$	−17893.6	−1.88074	−0.940368	+	0.340159i	$0.889519\pi$
$450$	65999.5	6.91388
$451$	−2210.90	−0.230837
$452$	−11642.3	−1.21152
$453$	18918.5	1.96218
$454$	26894.9	2.78026
$455$	0	0
$456$	13719.3	1.40892
$457$	6208.00	0.635444	0.317722	−	0.948184i	$-0.397082\pi$
$457$	6208.00	0.635444	0.317722	+	0.948184i	$0.397082\pi$
$458$	13313.0	1.35825
$459$	−3137.02	−0.319006
$460$	7280.01	0.737896
$461$	−7981.28	−0.806346	−0.403173	−	0.915124i	$-0.632093\pi$
$461$	−7981.28	−0.806346	−0.403173	+	0.915124i	$0.632093\pi$
$462$	0	0
$463$	−7495.19	−0.752334	−0.376167	−	0.926552i	$-0.622758\pi$
$463$	−7495.19	−0.752334	−0.376167	+	0.926552i	$0.622758\pi$
$464$	876.623	0.0877073
$465$	−15253.1	−1.52118
$466$	−290.484	−0.0288765
$467$	−1519.39	−0.150555	−0.0752773	−	0.997163i	$-0.523984\pi$
$467$	−1519.39	−0.150555	−0.0752773	+	0.997163i	$0.523984\pi$
$468$	25022.6	2.47152
$469$	0	0
$470$	26076.2	2.55916
$471$	2314.20	0.226397
$472$	10370.5	1.01131
$473$	−151.301	−0.0147079
$474$	−9593.90	−0.929667
$475$	−35837.7	−3.46178
$476$	0	0
$477$	12754.0	1.22425
$478$	−4898.75	−0.468753
$479$	16394.2	1.56382	0.781909	−	0.623393i	$-0.214246\pi$
$479$	16394.2	1.56382	0.781909	+	0.623393i	$0.214246\pi$
$480$	40380.1	3.83977
$481$	15800.7	1.49782
$482$	−23541.8	−2.22468
$483$	0	0
$484$	1421.64	0.133512
$485$	7686.45	0.719636
$486$	19270.1	1.79858
$487$	2275.01	0.211685	0.105843	−	0.994383i	$-0.466246\pi$
$487$	2275.01	0.211685	0.105843	+	0.994383i	$0.466246\pi$
$488$	1455.00	0.134968
$489$	7161.29	0.662259
$490$	0	0
$491$	−14629.2	−1.34462	−0.672308	−	0.740272i	$-0.734697\pi$
$491$	−14629.2	−1.34462	−0.672308	+	0.740272i	$0.734697\pi$
$492$	19515.0	1.78822
$493$	−1166.92	−0.106603
$494$	−22838.9	−2.08011
$495$	9999.68	0.907984
$496$	−1672.77	−0.151431
$497$	0	0
$498$	35035.1	3.15253
$499$	3534.01	0.317042	0.158521	−	0.987356i	$-0.449327\pi$
$499$	3534.01	0.317042	0.158521	+	0.987356i	$0.449327\pi$
$500$	60696.3	5.42884
$501$	−16240.1	−1.44822
$502$	−790.872	−0.0703155
$503$	9233.35	0.818479	0.409239	−	0.912427i	$-0.365794\pi$
$503$	9233.35	0.818479	0.409239	+	0.912427i	$0.365794\pi$
$504$	0	0
$505$	−32856.0	−2.89519
$506$	1375.86	0.120879
$507$	−3827.63	−0.335288
$508$	−7825.93	−0.683503
$509$	−11565.3	−1.00712	−0.503560	−	0.863960i	$-0.667977\pi$
$509$	−11565.3	−1.00712	−0.503560	+	0.863960i	$0.667977\pi$
$510$	−21472.9	−1.86439
$511$	0	0
$512$	7087.70	0.611787
$513$	11769.4	1.01292
$514$	−15539.3	−1.33348
$515$	−12299.5	−1.05239
$516$	1335.49	0.113937
$517$	2931.87	0.249407
$518$	0	0
$519$	32699.5	2.76560
$520$	18917.8	1.59539
$521$	2440.24	0.205200	0.102600	−	0.994723i	$-0.467284\pi$
$521$	2440.24	0.205200	0.102600	+	0.994723i	$0.467284\pi$
$522$	−8062.64	−0.676038
$523$	911.213	0.0761847	0.0380923	−	0.999274i	$-0.487872\pi$
$523$	911.213	0.0761847	0.0380923	+	0.999274i	$0.487872\pi$
$524$	−357.590	−0.0298119
$525$	0	0
$526$	25543.8	2.11742
$527$	2226.71	0.184055
$528$	1813.69	0.149490
$529$	−11374.8	−0.934892
$530$	30217.9	2.47657
$531$	25702.6	2.10056
$532$	0	0
$533$	−10366.5	−0.842445
$534$	−30845.6	−2.49966
$535$	−15279.4	−1.23474
$536$	10136.4	0.816838
$537$	25978.3	2.08761
$538$	−1045.30	−0.0837663
$539$	0	0
$540$	−30551.2	−2.43465
$541$	12277.5	0.975692	0.487846	−	0.872930i	$-0.337783\pi$
$541$	12277.5	0.975692	0.487846	+	0.872930i	$0.337783\pi$
$542$	5242.02	0.415432
$543$	5650.65	0.446580
$544$	−5894.84	−0.464594
$545$	−7507.22	−0.590044
$546$	0	0
$547$	−12539.2	−0.980141	−0.490071	−	0.871683i	$-0.663029\pi$
$547$	−12539.2	−0.980141	−0.490071	+	0.871683i	$0.663029\pi$
$548$	−33017.8	−2.57382
$549$	3606.11	0.280337
$550$	17581.6	1.36306
$551$	4378.01	0.338493
$552$	−3875.20	−0.298804
$553$	0	0
$554$	−16171.6	−1.24019
$555$	−55735.0	−4.26274
$556$	−36545.3	−2.78753
$557$	−14212.8	−1.08118	−0.540588	−	0.841287i	$-0.681798\pi$
$557$	−14212.8	−1.08118	−0.540588	+	0.841287i	$0.681798\pi$
$558$	15385.1	1.16721
$559$	−709.421	−0.0536768
$560$	0	0
$561$	−2414.30	−0.181697
$562$	14384.4	1.07966
$563$	−4446.83	−0.332880	−0.166440	−	0.986052i	$-0.553227\pi$
$563$	−4446.83	−0.332880	−0.166440	+	0.986052i	$0.553227\pi$
$564$	−25878.7	−1.93207
$565$	−21814.9	−1.62435
$566$	−36901.9	−2.74047
$567$	0	0
$568$	−7746.07	−0.572214
$569$	11258.8	0.829511	0.414756	−	0.909933i	$-0.363867\pi$
$569$	11258.8	0.829511	0.414756	+	0.909933i	$0.363867\pi$
$570$	80561.4	5.91991
$571$	−16450.6	−1.20567	−0.602835	−	0.797866i	$-0.705962\pi$
$571$	−16450.6	−1.20567	−0.602835	+	0.797866i	$0.705962\pi$
$572$	6665.78	0.487256
$573$	−22640.6	−1.65065
$574$	0	0
$575$	10122.8	0.734176
$576$	−34138.5	−2.46951
$577$	−10175.6	−0.734173	−0.367086	−	0.930187i	$-0.619645\pi$
$577$	−10175.6	−0.734173	−0.367086	+	0.930187i	$0.619645\pi$
$578$	−18698.6	−1.34561
$579$	−21915.4	−1.57301
$580$	−11364.5	−0.813597
$581$	0	0
$582$	−12822.4	−0.913237
$583$	3397.54	0.241358
$584$	4255.30	0.301516
$585$	46886.5	3.31371
$586$	8417.62	0.593394
$587$	5123.98	0.360289	0.180144	−	0.983640i	$-0.442344\pi$
$587$	5123.98	0.360289	0.180144	+	0.983640i	$0.442344\pi$
$588$	0	0
$589$	−8354.11	−0.584423
$590$	60896.7	4.24929
$591$	15532.3	1.08107
$592$	−6112.31	−0.424349
$593$	23816.7	1.64930	0.824650	−	0.565643i	$-0.191372\pi$
$593$	23816.7	1.64930	0.824650	+	0.565643i	$0.191372\pi$
$594$	−5773.93	−0.398834
$595$	0	0
$596$	22522.0	1.54788
$597$	−25778.4	−1.76724
$598$	6451.16	0.441150
$599$	11801.0	0.804965	0.402482	−	0.915428i	$-0.368147\pi$
$599$	11801.0	0.804965	0.402482	+	0.915428i	$0.368147\pi$
$600$	−49519.6	−3.36938
$601$	10944.5	0.742820	0.371410	−	0.928469i	$-0.378875\pi$
$601$	10944.5	0.742820	0.371410	+	0.928469i	$0.378875\pi$
$602$	0	0
$603$	25122.4	1.69662
$604$	−26896.9	−1.81195
$605$	2663.82	0.179007
$606$	54809.6	3.67407
$607$	1280.36	0.0856150	0.0428075	−	0.999083i	$-0.486370\pi$
$607$	1280.36	0.0856150	0.0428075	+	0.999083i	$0.486370\pi$
$608$	22116.1	1.47521
$609$	0	0
$610$	8543.91	0.567103
$611$	13747.0	0.910216
$612$	12885.2	0.851065
$613$	−11029.9	−0.726744	−0.363372	−	0.931644i	$-0.618375\pi$
$613$	−11029.9	−0.726744	−0.363372	+	0.931644i	$0.618375\pi$
$614$	−29313.9	−1.92673
$615$	36566.5	2.39757
$616$	0	0
$617$	−20861.3	−1.36117	−0.680586	−	0.732668i	$-0.738275\pi$
$617$	−20861.3	−1.36117	−0.680586	+	0.732668i	$0.738275\pi$
$618$	20517.7	1.33551
$619$	16877.9	1.09593	0.547966	−	0.836501i	$-0.315402\pi$
$619$	16877.9	1.09593	0.547966	+	0.836501i	$0.315402\pi$
$620$	21685.8	1.40471
$621$	−3324.42	−0.214822
$622$	23296.6	1.50178
$623$	0	0
$624$	8504.06	0.545568
$625$	68773.0	4.40147
$626$	−23725.4	−1.51479
$627$	9057.91	0.576935
$628$	−3290.17	−0.209064
$629$	8136.43	0.515772
$630$	0	0
$631$	−1332.58	−0.0840719	−0.0420359	−	0.999116i	$-0.513384\pi$
$631$	−1332.58	−0.0840719	−0.0420359	+	0.999116i	$0.513384\pi$
$632$	4352.42	0.273940
$633$	4303.19	0.270200
$634$	−25807.2	−1.61662
$635$	−14664.0	−0.916412
$636$	−29989.0	−1.86972
$637$	0	0
$638$	−2147.81	−0.133280
$639$	−19198.1	−1.18852
$640$	−41793.5	−2.58130
$641$	20472.2	1.26147	0.630736	−	0.775998i	$-0.282753\pi$
$641$	20472.2	1.26147	0.630736	+	0.775998i	$0.282753\pi$
$642$	25488.8	1.56692
$643$	−27140.3	−1.66455	−0.832276	−	0.554361i	$-0.812963\pi$
$643$	−27140.3	−1.66455	−0.832276	+	0.554361i	$0.812963\pi$
$644$	0	0
$645$	2502.39	0.152762
$646$	−11760.7	−0.716282
$647$	−13662.1	−0.830159	−0.415079	−	0.909785i	$-0.636246\pi$
$647$	−13662.1	−0.830159	−0.415079	+	0.909785i	$0.636246\pi$
$648$	−2312.67	−0.140201
$649$	6846.91	0.414121
$650$	82436.7	4.97451
$651$	0	0
$652$	−10181.4	−0.611556
$653$	1607.56	0.0963379	0.0481689	−	0.998839i	$-0.484661\pi$
$653$	1607.56	0.0963379	0.0481689	+	0.998839i	$0.484661\pi$
$654$	12523.4	0.748781
$655$	−670.040	−0.0399705
$656$	4010.15	0.238674
$657$	10546.5	0.626267
$658$	0	0
$659$	27361.6	1.61738	0.808692	−	0.588233i	$-0.200176\pi$
$659$	27361.6	1.61738	0.808692	+	0.588233i	$0.200176\pi$
$660$	−23512.7	−1.38671
$661$	5117.29	0.301119	0.150559	−	0.988601i	$-0.451893\pi$
$661$	5117.29	0.301119	0.150559	+	0.988601i	$0.451893\pi$
$662$	−6072.76	−0.356533
$663$	−11320.2	−0.663107
$664$	−15894.2	−0.928939
$665$	0	0
$666$	56217.3	3.27083
$667$	−1236.63	−0.0717877
$668$	23089.0	1.33734
$669$	17363.2	1.00344
$670$	59522.1	3.43215
$671$	960.633	0.0552680
$672$	0	0
$673$	11605.6	0.664729	0.332365	−	0.943151i	$-0.392154\pi$
$673$	11605.6	0.664729	0.332365	+	0.943151i	$0.392154\pi$
$674$	−14948.5	−0.854292
$675$	−42481.3	−2.42238
$676$	5441.85	0.309618
$677$	−32514.6	−1.84584	−0.922922	−	0.384986i	$-0.874206\pi$
$677$	−32514.6	−1.84584	−0.922922	+	0.384986i	$0.874206\pi$
$678$	36391.1	2.06134
$679$	0	0
$680$	9741.54	0.549369
$681$	−50013.1	−2.81425
$682$	4098.44	0.230113
$683$	7201.06	0.403427	0.201714	−	0.979445i	$-0.435349\pi$
$683$	7201.06	0.403427	0.201714	+	0.979445i	$0.435349\pi$
$684$	−48342.1	−2.70235
$685$	−61867.6	−3.45086
$686$	0	0
$687$	−24756.6	−1.37485
$688$	274.431	0.0152072
$689$	15930.4	0.880841
$690$	−22755.7	−1.25550
$691$	32357.7	1.78140	0.890698	−	0.454596i	$-0.150216\pi$
$691$	32357.7	1.78140	0.890698	+	0.454596i	$0.150216\pi$
$692$	−46489.7	−2.55386
$693$	0	0
$694$	13265.2	0.725562
$695$	−68477.3	−3.73740
$696$	6049.42	0.329458
$697$	−5338.13	−0.290095
$698$	−5715.31	−0.309925
$699$	540.178	0.0292295
$700$	0	0
$701$	11077.3	0.596838	0.298419	−	0.954435i	$-0.403541\pi$
$701$	11077.3	0.596838	0.298419	+	0.954435i	$0.403541\pi$
$702$	−27072.8	−1.45555
$703$	−30526.0	−1.63771
$704$	−9094.16	−0.486859
$705$	−48490.6	−2.59044
$706$	−37407.7	−1.99413
$707$	0	0
$708$	−60435.6	−3.20806
$709$	−28594.0	−1.51463	−0.757314	−	0.653051i	$-0.773489\pi$
$709$	−28594.0	−1.51463	−0.757314	+	0.653051i	$0.773489\pi$
$710$	−45485.8	−2.40430
$711$	10787.2	0.568990
$712$	13993.6	0.736563
$713$	2359.73	0.123945
$714$	0	0
$715$	12490.1	0.653292
$716$	−36934.1	−1.92778
$717$	9109.61	0.474483
$718$	33256.5	1.72858
$719$	−18240.5	−0.946114	−0.473057	−	0.881032i	$-0.656850\pi$
$719$	−18240.5	−0.946114	−0.473057	+	0.881032i	$0.656850\pi$
$720$	−18137.5	−0.938811
$721$	0	0
$722$	13642.0	0.703190
$723$	43777.7	2.25188
$724$	−8033.68	−0.412389
$725$	−15802.4	−0.809496
$726$	−4443.72	−0.227165
$727$	9792.53	0.499566	0.249783	−	0.968302i	$-0.419641\pi$
$727$	9792.53	0.499566	0.249783	+	0.968302i	$0.419641\pi$
$728$	0	0
$729$	−32086.4	−1.63016
$730$	24987.6	1.26690
$731$	−365.309	−0.0184835
$732$	−8479.22	−0.428143
$733$	22723.8	1.14505	0.572525	−	0.819887i	$-0.305964\pi$
$733$	22723.8	1.14505	0.572525	+	0.819887i	$0.305964\pi$
$734$	−38168.3	−1.91937
$735$	0	0
$736$	−6246.98	−0.312863
$737$	6692.35	0.334485
$738$	−36882.9	−1.83967
$739$	−24063.6	−1.19783	−0.598913	−	0.800814i	$-0.704401\pi$
$739$	−24063.6	−1.19783	−0.598913	+	0.800814i	$0.704401\pi$
$740$	79240.0	3.93638
$741$	42470.8	2.10554
$742$	0	0
$743$	−29211.6	−1.44235	−0.721177	−	0.692751i	$-0.756399\pi$
$743$	−29211.6	−1.44235	−0.721177	+	0.692751i	$0.756399\pi$
$744$	−11543.5	−0.568824
$745$	42200.9	2.07533
$746$	52585.6	2.58083
$747$	−39392.8	−1.92946
$748$	3432.48	0.167786
$749$	0	0
$750$	−189723.	−9.23692
$751$	1880.93	0.0913931	0.0456965	−	0.998955i	$-0.485449\pi$
$751$	1880.93	0.0913931	0.0456965	+	0.998955i	$0.485449\pi$
$752$	−5317.84	−0.257874
$753$	1470.69	0.0711751
$754$	−10070.6	−0.486408
$755$	−50398.4	−2.42939
$756$	0	0
$757$	−36218.7	−1.73896	−0.869480	−	0.493968i	$-0.835546\pi$
$757$	−36218.7	−1.73896	−0.869480	+	0.493968i	$0.835546\pi$
$758$	22470.5	1.07674
$759$	−2558.53	−0.122357
$760$	−36548.0	−1.74439
$761$	−36966.4	−1.76088	−0.880441	−	0.474156i	$-0.842753\pi$
$761$	−36966.4	−1.76088	−0.880441	+	0.474156i	$0.842753\pi$
$762$	24462.1	1.16295
$763$	0	0
$764$	32188.7	1.52428
$765$	24143.8	1.14107
$766$	−28695.2	−1.35353
$767$	32103.8	1.51135
$768$	15061.8	0.707678
$769$	38975.5	1.82769	0.913845	−	0.406062i	$-0.133098\pi$
$769$	38975.5	1.82769	0.913845	+	0.406062i	$0.133098\pi$
$770$	0	0
$771$	28896.5	1.34978
$772$	31157.7	1.45258
$773$	−27341.9	−1.27221	−0.636105	−	0.771603i	$-0.719455\pi$
$773$	−27341.9	−1.27221	−0.636105	+	0.771603i	$0.719455\pi$
$774$	−2524.05	−0.117216
$775$	30154.0	1.39763
$776$	5817.07	0.269099
$777$	0	0
$778$	55303.3	2.54848
$779$	20027.4	0.921125
$780$	−110246.	−5.06084
$781$	−5114.19	−0.234315
$782$	3321.96	0.151909
$783$	5189.61	0.236860
$784$	0	0
$785$	−6165.00	−0.280303
$786$	1117.75	0.0507235
$787$	18268.5	0.827446	0.413723	−	0.910403i	$-0.364228\pi$
$787$	18268.5	0.827446	0.413723	+	0.910403i	$0.364228\pi$
$788$	−22082.7	−0.998302
$789$	−47500.7	−2.14331
$790$	25558.0	1.15103
$791$	0	0
$792$	7567.71	0.339529
$793$	4504.22	0.201702
$794$	2751.75	0.122993
$795$	−56192.4	−2.50684
$796$	36649.9	1.63194
$797$	−12717.6	−0.565219	−0.282610	−	0.959235i	$-0.591200\pi$
$797$	−12717.6	−0.565219	−0.282610	+	0.959235i	$0.591200\pi$
$798$	0	0
$799$	7078.86	0.313432
$800$	−79827.6	−3.52792
$801$	34682.2	1.52988
$802$	43248.4	1.90418
$803$	2809.48	0.123467
$804$	−59071.4	−2.59115
$805$	0	0
$806$	19216.8	0.839804
$807$	1943.82	0.0847904
$808$	−24865.3	−1.08262
$809$	12502.0	0.543322	0.271661	−	0.962393i	$-0.412427\pi$
$809$	12502.0	0.543322	0.271661	+	0.962393i	$0.412427\pi$
$810$	−13580.3	−0.589090
$811$	23431.3	1.01453	0.507264	−	0.861791i	$-0.330657\pi$
$811$	23431.3	1.01453	0.507264	+	0.861791i	$0.330657\pi$
$812$	0	0
$813$	−9747.95	−0.420511
$814$	14975.7	0.644839
$815$	−19077.6	−0.819948
$816$	4379.08	0.187866
$817$	1370.56	0.0586899
$818$	−20537.6	−0.877850
$819$	0	0
$820$	−51987.6	−2.21401
$821$	33116.5	1.40776	0.703881	−	0.710317i	$-0.251449\pi$
$821$	33116.5	1.40776	0.703881	+	0.710317i	$0.251449\pi$
$822$	103206.	4.37923
$823$	−6383.39	−0.270366	−0.135183	−	0.990821i	$-0.543162\pi$
$823$	−6383.39	−0.270366	−0.135183	+	0.990821i	$0.543162\pi$
$824$	−9308.20	−0.393527
$825$	−32694.4	−1.37972
$826$	0	0
$827$	−27701.3	−1.16477	−0.582386	−	0.812912i	$-0.697881\pi$
$827$	−27701.3	−1.16477	−0.582386	+	0.812912i	$0.697881\pi$
$828$	13654.9	0.573116
$829$	13160.2	0.551353	0.275676	−	0.961251i	$-0.411098\pi$
$829$	13160.2	0.551353	0.275676	+	0.961251i	$0.411098\pi$
$830$	−93332.7	−3.90316
$831$	30072.3	1.25535
$832$	−42640.7	−1.77680
$833$	0	0
$834$	114232.	4.74286
$835$	43263.4	1.79305
$836$	−12877.9	−0.532763
$837$	−9902.80	−0.408950
$838$	−828.483	−0.0341521
$839$	24842.9	1.02226	0.511128	−	0.859505i	$-0.329228\pi$
$839$	24842.9	1.02226	0.511128	+	0.859505i	$0.329228\pi$
$840$	0	0
$841$	−22458.6	−0.920848
$842$	11866.8	0.485696
$843$	−26748.8	−1.09286
$844$	−6117.95	−0.249513
$845$	10196.7	0.415123
$846$	48910.2	1.98767
$847$	0	0
$848$	−6162.48	−0.249552
$849$	68622.0	2.77397
$850$	42450.0	1.71297
$851$	8622.47	0.347326
$852$	45141.4	1.81516
$853$	10131.6	0.406681	0.203340	−	0.979108i	$-0.434820\pi$
$853$	10131.6	0.406681	0.203340	+	0.979108i	$0.434820\pi$
$854$	0	0
$855$	−90581.8	−3.62320
$856$	−11563.4	−0.461716
$857$	−10115.6	−0.403199	−0.201599	−	0.979468i	$-0.564614\pi$
$857$	−10115.6	−0.403199	−0.201599	+	0.979468i	$0.564614\pi$
$858$	−20835.7	−0.829044
$859$	27491.4	1.09196	0.545980	−	0.837798i	$-0.316157\pi$
$859$	27491.4	1.09196	0.545980	+	0.837798i	$0.316157\pi$
$860$	−3557.72	−0.141066
$861$	0	0
$862$	−55616.1	−2.19755
$863$	−117.276	−0.00462588	−0.00231294	−	0.999997i	$-0.500736\pi$
$863$	−117.276	−0.00462588	−0.00231294	+	0.999997i	$0.500736\pi$
$864$	26216.0	1.03228
$865$	−87110.8	−3.42411
$866$	73997.8	2.90364
$867$	34771.5	1.36206
$868$	0	0
$869$	2873.60	0.112175
$870$	35522.9	1.38430
$871$	31379.1	1.22071
$872$	−5681.43	−0.220639
$873$	14417.2	0.558933
$874$	−12463.2	−0.482351
$875$	0	0
$876$	−24798.4	−0.956462
$877$	11597.4	0.446543	0.223271	−	0.974756i	$-0.428326\pi$
$877$	11597.4	0.446543	0.223271	+	0.974756i	$0.428326\pi$
$878$	−26812.3	−1.03061
$879$	−15653.2	−0.600648
$880$	−4831.65	−0.185085
$881$	−7524.18	−0.287737	−0.143868	−	0.989597i	$-0.545954\pi$
$881$	−7524.18	−0.287737	−0.143868	+	0.989597i	$0.545954\pi$
$882$	0	0
$883$	13467.4	0.513266	0.256633	−	0.966509i	$-0.417387\pi$
$883$	13467.4	0.513266	0.256633	+	0.966509i	$0.417387\pi$
$884$	16094.2	0.612339
$885$	−113242.	−4.30124
$886$	28086.2	1.06498
$887$	12955.7	0.490427	0.245214	−	0.969469i	$-0.421142\pi$
$887$	12955.7	0.490427	0.245214	+	0.969469i	$0.421142\pi$
$888$	−42180.0	−1.59400
$889$	0	0
$890$	82172.1	3.09485
$891$	−1526.90	−0.0574107
$892$	−24685.7	−0.926612
$893$	−26558.2	−0.995226
$894$	−70398.6	−2.63365
$895$	−69205.8	−2.58469
$896$	0	0
$897$	−11996.4	−0.446543
$898$	−79518.9	−2.95499
$899$	−3683.68	−0.136660
$900$	174490.	6.46260
$901$	8203.20	0.303317
$902$	−9825.25	−0.362688
$903$	0	0
$904$	−16509.4	−0.607405
$905$	−15053.2	−0.552913
$906$	84073.5	3.08295
$907$	47843.5	1.75151	0.875753	−	0.482759i	$-0.160365\pi$
$907$	47843.5	1.75151	0.875753	+	0.482759i	$0.160365\pi$
$908$	71104.9	2.59879
$909$	−61626.9	−2.24866
$910$	0	0
$911$	16969.8	0.617162	0.308581	−	0.951198i	$-0.400146\pi$
$911$	16969.8	0.617162	0.308581	+	0.951198i	$0.400146\pi$
$912$	−16429.3	−0.596522
$913$	−10493.8	−0.380389
$914$	27588.3	0.998402
$915$	−15888.1	−0.574036
$916$	35197.1	1.26959
$917$	0	0
$918$	−13940.9	−0.501218
$919$	12095.2	0.434149	0.217075	−	0.976155i	$-0.430348\pi$
$919$	12095.2	0.434149	0.217075	+	0.976155i	$0.430348\pi$
$920$	10323.5	0.369951
$921$	54511.4	1.95029
$922$	−35468.8	−1.26692
$923$	−23979.4	−0.855138
$924$	0	0
$925$	110183.	3.91653
$926$	−33308.6	−1.18206
$927$	−23069.8	−0.817379
$928$	9751.91	0.344959
$929$	44544.3	1.57314	0.786571	−	0.617499i	$-0.211854\pi$
$929$	44544.3	1.57314	0.786571	+	0.617499i	$0.211854\pi$
$930$	−67784.8	−2.39005
$931$	0	0
$932$	−767.985	−0.0269916
$933$	−43321.8	−1.52014
$934$	−6752.16	−0.236550
$935$	6431.66	0.224960
$936$	35483.5	1.23912
$937$	49265.8	1.71766	0.858828	−	0.512264i	$-0.171193\pi$
$937$	49265.8	1.71766	0.858828	+	0.512264i	$0.171193\pi$
$938$	0	0
$939$	44119.2	1.53331
$940$	68940.4	2.39211
$941$	18403.1	0.637538	0.318769	−	0.947832i	$-0.396731\pi$
$941$	18403.1	0.637538	0.318769	+	0.947832i	$0.396731\pi$
$942$	10284.3	0.355712
$943$	−5657.01	−0.195353
$944$	−12419.0	−0.428181
$945$	0	0
$946$	−672.381	−0.0231089
$947$	17689.3	0.606996	0.303498	−	0.952832i	$-0.401845\pi$
$947$	17689.3	0.606996	0.303498	+	0.952832i	$0.401845\pi$
$948$	−25364.4	−0.868986
$949$	13173.1	0.450597
$950$	−159262.	−5.43911
$951$	47990.5	1.63638
$952$	0	0
$953$	−5298.19	−0.180090	−0.0900448	−	0.995938i	$-0.528701\pi$
$953$	−5298.19	−0.180090	−0.0900448	+	0.995938i	$0.528701\pi$
$954$	56678.7	1.92352
$955$	60314.1	2.04368
$956$	−12951.4	−0.438156
$957$	3994.01	0.134909
$958$	72855.6	2.45705
$959$	0	0
$960$	150410.	5.05672
$961$	−22761.8	−0.764050
$962$	70218.3	2.35336
$963$	−28659.1	−0.959011
$964$	−62239.9	−2.07947
$965$	58382.2	1.94755
$966$	0	0
$967$	33990.6	1.13037	0.565184	−	0.824965i	$-0.308805\pi$
$967$	33990.6	1.13037	0.565184	+	0.824965i	$0.308805\pi$
$968$	2015.96	0.0669375
$969$	21869.9	0.725039
$970$	34158.5	1.13068
$971$	−41991.0	−1.38780	−0.693900	−	0.720071i	$-0.744109\pi$
$971$	−41991.0	−1.38780	−0.693900	+	0.720071i	$0.744109\pi$
$972$	50946.5	1.68118
$973$	0	0
$974$	10110.1	0.332597
$975$	−153297.	−5.03533
$976$	−1742.40	−0.0571444
$977$	−31233.4	−1.02277	−0.511384	−	0.859352i	$-0.670867\pi$
$977$	−31233.4	−1.02277	−0.511384	+	0.859352i	$0.670867\pi$
$978$	31824.7	1.04053
$979$	9239.00	0.301614
$980$	0	0
$981$	−14081.0	−0.458281
$982$	−65012.0	−2.11264
$983$	45702.5	1.48289	0.741447	−	0.671012i	$-0.234140\pi$
$983$	45702.5	1.48289	0.741447	+	0.671012i	$0.234140\pi$
$984$	27673.4	0.896540
$985$	−41377.7	−1.33848
$986$	−5185.78	−0.167494
$987$	0	0
$988$	−60381.8	−1.94433
$989$	−387.132	−0.0124470
$990$	44438.5	1.42661
$991$	−4310.72	−0.138178	−0.0690890	−	0.997611i	$-0.522009\pi$
$991$	−4310.72	−0.138178	−0.0690890	+	0.997611i	$0.522009\pi$
$992$	−18608.6	−0.595587
$993$	11292.8	0.360892
$994$	0	0
$995$	68673.3	2.18803
$996$	92626.0	2.94675
$997$	6103.28	0.193875	0.0969373	−	0.995290i	$-0.469095\pi$
$997$	6103.28	0.193875	0.0969373	+	0.995290i	$0.469095\pi$
$998$	15705.1	0.498133
$999$	−36184.9	−1.14599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.4.a.h.1.4		5
7.6	odd	2		77.4.a.e.1.4	✓	5
21.20	even	2		693.4.a.o.1.2		5
28.27	even	2		1232.4.a.y.1.1		5
35.34	odd	2		1925.4.a.r.1.2		5
77.76	even	2		847.4.a.f.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.4	✓	5	7.6	odd	2
539.4.a.h.1.4		5	1.1	even	1	trivial
693.4.a.o.1.2		5	21.20	even	2
847.4.a.f.1.2		5	77.76	even	2
1232.4.a.y.1.1		5	28.27	even	2
1925.4.a.r.1.2		5	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q+4.44399 q^{2} -8.26395 q^{3} +11.7491 q^{4} +22.0150 q^{5} -36.7249 q^{6} +16.6609 q^{8} +41.2928 q^{9} +97.8345 q^{10} +11.0000 q^{11} -97.0937 q^{12} +51.5769 q^{13} -181.931 q^{15} -19.9519 q^{16} +26.5590 q^{17} +183.505 q^{18} -99.6432 q^{19} +258.656 q^{20} +48.8839 q^{22} +28.1455 q^{23} -137.684 q^{24} +359.660 q^{25} +229.207 q^{26} -118.115 q^{27} -43.9369 q^{29} -808.499 q^{30} +83.8402 q^{31} -221.953 q^{32} -90.9034 q^{33} +118.028 q^{34} +485.152 q^{36} +306.353 q^{37} -442.814 q^{38} -426.228 q^{39} +366.789 q^{40} -200.991 q^{41} -13.7546 q^{43} +129.240 q^{44} +909.062 q^{45} +125.079 q^{46} +266.533 q^{47} +164.881 q^{48} +1598.33 q^{50} -219.482 q^{51} +605.980 q^{52} +308.867 q^{53} -524.903 q^{54} +242.165 q^{55} +823.446 q^{57} -195.255 q^{58} +622.446 q^{59} -2137.52 q^{60} +87.3303 q^{61} +372.585 q^{62} -826.742 q^{64} +1135.46 q^{65} -403.974 q^{66} +608.395 q^{67} +312.044 q^{68} -232.593 q^{69} -464.926 q^{71} +687.974 q^{72} +255.407 q^{73} +1361.43 q^{74} -2972.21 q^{75} -1170.72 q^{76} -1894.16 q^{78} +261.237 q^{79} -439.240 q^{80} -138.809 q^{81} -893.204 q^{82} -953.986 q^{83} +584.696 q^{85} -61.1255 q^{86} +363.092 q^{87} +183.269 q^{88} +839.910 q^{89} +4039.86 q^{90} +330.684 q^{92} -692.851 q^{93} +1184.47 q^{94} -2193.64 q^{95} +1834.21 q^{96} +349.146 q^{97} +454.221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} - 2 q^{3} + 45 q^{4} + 24 q^{5} - 4 q^{6} + 57 q^{8} + 63 q^{9} + 10 q^{10} + 55 q^{11} - 24 q^{12} + 50 q^{13} - 146 q^{15} + 433 q^{16} - 222 q^{17} + 245 q^{18} - 160 q^{19} + 430 q^{20}+ \cdots + 693 q^{99}+O(q^{100}) \) 5 * q + q^2 - 2 * q^3 + 45 * q^4 + 24 * q^5 - 4 * q^6 + 57 * q^8 + 63 * q^9 + 10 * q^10 + 55 * q^11 - 24 * q^12 + 50 * q^13 - 146 * q^15 + 433 * q^16 - 222 * q^17 + 245 * q^18 - 160 * q^19 + 430 * q^20 + 11 * q^22 + 54 * q^23 + 220 * q^24 + 125 * q^25 + 1026 * q^26 - 110 * q^27 + 14 * q^29 - 1080 * q^30 + 34 * q^31 - 583 * q^32 - 22 * q^33 + 750 * q^34 - 411 * q^36 + 1044 * q^37 + 156 * q^38 - 124 * q^39 - 1158 * q^40 + 114 * q^41 + 672 * q^43 + 495 * q^44 + 530 * q^45 - 1224 * q^46 + 292 * q^47 + 1652 * q^48 + 1143 * q^50 + 768 * q^51 + 914 * q^52 - 710 * q^53 - 1608 * q^54 + 264 * q^55 + 2012 * q^57 - 810 * q^58 - 270 * q^59 - 3068 * q^60 - 138 * q^61 + 480 * q^62 + 3001 * q^64 - 196 * q^65 - 44 * q^66 + 1942 * q^67 - 3130 * q^68 + 1306 * q^69 - 278 * q^71 + 3565 * q^72 + 338 * q^73 + 462 * q^74 - 3960 * q^75 + 1332 * q^76 + 280 * q^78 + 576 * q^79 + 1602 * q^80 - 1439 * q^81 - 1386 * q^82 - 1644 * q^83 + 360 * q^85 + 2020 * q^86 + 1956 * q^87 + 627 * q^88 + 3656 * q^89 + 6162 * q^90 - 748 * q^92 - 1442 * q^93 - 976 * q^94 - 1276 * q^95 + 4508 * q^96 - 692 * q^97 + 693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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